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Error Analysis of a Decoupled, Linear Stabilization Scheme for the Cahn–Hilliard Model of Two-Phase Incompressible Flows

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Abstract

Here, we carry out rigorous error analysis for a first-order in time, linear, fully decoupled and energy stable scheme for solving the Cahn–Hilliard phase-field model of two-phase incompressible flows, namely Cahn–Hilliard–Navier–Stokes problem (Shen and Yang, SIAM J Numer Anal, 2015). The error estimates are for phase field variable, chemical potential, velocity and further the pressure in \(L^2\) norm and \(L^{\infty }\) norm. The scheme combines the projection method, the explicit stabilizing decoupling technique, and the linear stabilization approach together. We further derive the boundness of numerical solution in \(L^\infty \) norm with the mathematical deduction, and deal with the complex splitting error arising from the decoupling technique. Optimal error estimates are derived for the semi-discrete-in-time scheme.

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Acknowledgements

Z. Xu was partially supported by China Scholarship Council (No. 201706040140). X. Yang was partially supported by NSF DMS-1720212 and 1818783. H. Zhang was partially supported by NSFC-11471046 and NSFC-11571045. This paper is dedicated to the memory of Professor Hui Zhang, one of the co-authors, who passed suddenly away on Feb 26, 2020. There are no words to express our sorrow for his loss as close friends. The co-author, Z. Xu wishes to express the highest respect and gratitude to her passed thesis advisor, Professor Hui Zhang, for his help on the research and life. His rigorous attitude and optimistic spirit always inspire us. He will be missed by us forever.

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  1. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, People’s Republic of China

    Zhen Xu

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Hui Zhang

  4. Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, People’s Republic of China

    Hui Zhang

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  1. Zhen Xu
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  2. Xiaofeng Yang
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  3. Hui Zhang
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Xu, Z., Yang, X. & Zhang, H. Error Analysis of a Decoupled, Linear Stabilization Scheme for the Cahn–Hilliard Model of Two-Phase Incompressible Flows. J Sci Comput 83, 57 (2020). https://doi.org/10.1007/s10915-020-01241-w

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